Quadratic twists of abelian varieties with real multiplication

2017/10/11 by Shnidman, Ari
#11G10 #14G05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1710.04086

Abstract

Let F be a totally real number field and A/F a principally polarized abelian variety with real multiplication by the ring of integers O of a totally real field. Assuming A admits an O-linear 3-isogeny over F, we prove that a positive proportion of the quadratic twists A_d have rank 0. We also prove that a positive proportion of A_d have rank dim A, assuming the Tate-Shafarevich groups are finite. If A is the Jacobian of a hyperelliptic curve C, we deduce that a positive proportion of twists C_d have no rational points other than those fixed by the hyperelliptic involution.

Related

• Constructing Jacobians of rank 1
• Counting the number of the twists of certain polarized abelian varieties
• Torsion and twists of abelian varieties
• Principal polarizations on products of abelian varieties over finite fields
• Geometrically simple counterexamples to a local-global principle for quadratic twists